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This page shows questions in the Alg II Reasoning public release module at MSDE. Algebra 2
"Alg II Reasoning"

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Expand All Official Practice Test

This is a question with 2 parts, including a test question that allows you to enter extended text in your response.

Claim: There is no quadratic equation with real coefficients for which x equals negative 5 I is a solution.

Part A

Show that the claim is not correct by providing an example of a quadratic equation for which x equals negative 5 I is a solution. Include any other solutions to your equation in your answer.

Enter your answer and your explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Part B

Is it possible for a quadratic equation with real coefficients to have x equals negative 5 I as its only solution? Justify your answer.

Enter your answer and your justification in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

There is no alt-text. (37 files were searched.)

Holistic Rubric for 4-Point Reasoning Constructed Response Items

Sample Top Score Response

Part A

A quadratic equation with real coefficients that has x = − 5 i as a solution must also have x = 5 i as a solution. One such equation is ( x − 5 i ) ( x + 5 i ) = 0 , which is equivalent to x 2 + 25 = 0 .

Part B

There is no quadratic equation with real coefficients that has x = − 5 i as its only solution. If the only solution is x = − 5 i , then the quadratic equation is a multiple of ( x + 5 i ) 2 = 0 , which is equivalent to x 2 + 10 i x − 25 = 0 , and that equation cannot be equivalent to one with real coefficients because 10 i − 25 is not a real number.

Scoring

Points

Sample Characteristics

4 Points

A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning.

The response may:

  • be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • demonstrate a thorough understanding of the mathematics.
  • contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.

3 Points

A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning.

The response may:

  • be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.

2 Points

A two-point response for reasoning items provides evidence of partially correct mathematical reasoning.

The response may:

  • display an incomplete reasoning process.
  • contain mathematical flaws.

1 Point

A one-point response for reasoning items provides limited evidence of correct mathematical reasoning.

The response may:

  • demonstrate the beginning of a valid chain of reasoning.
  • reflect a lack of essential understanding of the underlying mathematical concepts.
  • contain the correct solution, but work is limited or missing.
  • contain errors in the fundamental mathematical procedures or reasoning.
  • contain omissions or irregularities that lead to an inadequate solution.

0 Point

A zero-point response is completely incorrect, incoherent or irrelevant.

This is a multiple choice question that allows you to select only one option.

Consider the polynomial function P, defined by P of x equals x cubed plus C X squared plus x plus 2, where c is a constant, and consider the graph shown in the xy-coordinate plane.

Figure. The figure shows the graph of a curve in the x y plane. The numbers negative 4, through 4, are indicated along both the x and y axes. The curve moves through all four quadrants of the plane, crossing the x axis three times, and crossing the y axis one time. From left to right, the curve moves upward, and to the right, through quadrant 3, crosses the x axis at negative 2, and continues upward, and to the right, in quadrant 2. The curve then reaches a relative maximum point, and begins to move downward, and to the right, crosses the y axis at 2, continues downward, and to the right, through quadrant 1, and crosses the axis at 1. The curve continues downward, and to the right, in quadrant 4, reaches a relative minimum, and then moves upward, and to the right, crosses the x axis at 3, and continues upward and to the right through quadrant 1. End figure description.

If P of negative 2 equals 0, can the graph shown be the graph of y equals P of x?

  1. A.
    Yes, because c would equal negative 2 and the points with coordinates 1, 0 and 3, 0 would be on the graph of y equals p of x.
  2. B.
    Yes, because c would equal 2, and the points with coordinates 1, 0 and 3, 0 would be on the graph of y equals p of x.
  3. C.
    No, because c would equal negative 2, and the points with coordinates negative 2, 0 and (3,0) would not be on the graph of y=P(x).
  4. D.
    No, because c would equal 2, and the points with coordinates 1, 0 and (3,0) would not be on the graph of y=P(x).
i4

Correct.i1

Incorrect conclusion and supporting factsi2

Incorrect conclusion and some incorrect supporting factsi3

Correct conclusion but incorrect supporting facts

This is a matching question that allows you to match elements from one list with those on another list.

Claim: x3≤x, where x is a real number.

Determine whether the claim is true for values of x in each interval.

Select only one box per row.

Interval The Claim is True The Claim is False
x>1
0<x<1
−1<x<0
x<−1

This is a test question that allows you to enter extended text in your response.

Consider the functions P and Q, defined as shown.Equation 1: P of x, equals x squared plus 7 x, minus 14. Equation 2: Q of x, equals negative 3 x plus 10.

In the xy-coordinate plane, what are the coordinates of the points at which the graphs of the equations y=P(x) and y=Q(x) intersect?

Explain how you determined your answer.

Enter your answer and your explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

There is no alt-text. (37 files were searched.)

Holistic Rubric for 4-Point Reasoning Constructed Response Items

Sample Top Score Response

The two graphs intersect at the points (x,y)=(−12,46) and (x,y)=(2,4).

The x coordinates of the points of intersection of the graph are those points for which P(x)=Q(x).

P(x)=Q(x)x2+7x-14=-3x+10x2+10x-24=0(x+12)(x-2)=0

x=−12 or x=2

If x=−12, then y=(−3)(−12)+10=36+10=46

If x=2, then y=(−3)(2)+10=−6+10=4

Therefore, the points of intersection are (x,y)=(−12,46) and (x,y)=(2,4).

Points

Sample Characteristics

4 Points

A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning.

The response may:

  • be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • demonstrate a thorough understanding of the mathematics.
  • contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.

3 Points

A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning.

The response may:

  • be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.

2 Points

A two-point response for reasoning items provides evidence of partially correct mathematical reasoning.

The response may:

  • display an incomplete reasoning process.
  • contain mathematical flaws.

1 Point

A one-point response for reasoning items provides limited evidence of correct mathematical reasoning.

The response may:

  • demonstrate the beginning of a valid chain of reasoning.
  • reflect a lack of essential understanding of the underlying mathematical concepts.
  • contain the correct solution, but work is limited or missing.
  • contain errors in the fundamental mathematical procedures or reasoning.
  • contain omissions or irregularities that lead to an inadequate solution.

0 Point

A zero-point response is completely incorrect, incoherent or irrelevant.

This is a fill in the blank question that allows you to enter only numbers.

Claim: For every value of the constant c, the equation x4+(1−c2)x2−c2=0 has at least two distinct real solutions.

What value of c refutes the claim?

Enter your answer in the space provided.

c=

This is a multiple choice question that allows you to select only one option.

Consider the equation a=x+b, where a and b represent real numbers.

Which statement is true about the number of real solutions to the equation?

  1. A.
    The number of real solutions depends only on the value of a.
  2. B.
    The number of real solutions depends only on the value of b.
  3. C.
    The number of real solutions depends on the relationship between a and b.
  4. D.
    The values of a and b do not affect the number of real solutions.
i1

KEY i2

Student thinks b affects the number of solutions. i3

Student thinks the relationship between a and b affects the number of solutions. i4

Student thinks the number of solutions does not depend on a and b .